Proper time - Wikipedia

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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples in special relativity</span> </div> </a> <button aria-controls="toc-Examples_in_special_relativity-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples in special relativity subsection</span> </button> <ul id="toc-Examples_in_special_relativity-sublist" class="vector-toc-list"> <li id="toc-Example_1:_The_twin_"paradox"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_1:_The_twin_"paradox""> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Example 1: The twin "paradox"</span> </div> </a> <ul id="toc-Example_1:_The_twin_"paradox"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2:_The_rotating_disk" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" 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class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_3:_The_rotating_disk_(again)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Example 3: The rotating disk (again)</span> </div> </a> <ul id="toc-Example_3:_The_rotating_disk_(again)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_4:_The_Schwarzschild_solution_–_time_on_the_Earth" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_4:_The_Schwarzschild_solution_–_time_on_the_Earth"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Example 4: The Schwarzschild solution – time on the Earth</span> </div> </a> <ul id="toc-Example_4:_The_Schwarzschild_solution_–_time_on_the_Earth-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Proper time</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%95%E0%A7%83%E0%A6%A4_%E0%A6%B8%E0%A6%AE%E0%A6%AF%E0%A6%BC" title="প্রকৃত সময় – Bangla" lang="bn" hreflang="bn" data-title="প্রকৃত সময়" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Temps_propi" title="Temps propi – Catalan" lang="ca" hreflang="ca" data-title="Temps propi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%C4%83%D0%B9_%D0%B2%C4%83%D1%85%C4%83%D1%82" title="Хăй вăхăт – Chuvash" lang="cv" hreflang="cv" data-title="Хăй вăхăт" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlastn%C3%AD_%C4%8Das" title="Vlastní čas – Czech" lang="cs" hreflang="cs" data-title="Vlastní čas" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Egentid" title="Egentid – Danish" lang="da" hreflang="da" data-title="Egentid" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Zeitdilatation#Eigenzeit" title="Zeitdilatation – German" lang="de" hreflang="de" data-title="Zeitdilatation" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Tiempo_propio" title="Tiempo propio – Spanish" lang="es" hreflang="es" data-title="Tiempo propio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%D9%85%D8%A7%D9%86_%D9%88%DB%8C%DA%98%D9%87" title="زمان ویژه – Persian" lang="fa" hreflang="fa" data-title="زمان ویژه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Temps_propre" title="Temps propre – French" lang="fr" hreflang="fr" data-title="Temps propre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B3%A0%EC%9C%A0_%EC%8B%9C%EA%B0%84" title="고유 시간 – Korean" lang="ko" hreflang="ko" data-title="고유 시간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Tempo_proprio" title="Tempo proprio – Italian" lang="it" hreflang="it" data-title="Tempo proprio" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%96%D7%9E%D7%9F_%D7%A2%D7%A6%D7%9E%D7%99" title="זמן עצמי – Hebrew" lang="he" hreflang="he" data-title="זמן עצמי" data-language-autonym="עברית" data-language-local-name="Hebrew" 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by a clock that passes through both events</div> <p>In <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</a>, <b>proper time</b> (from Latin, meaning <i>own time</i>) along a <a href="/wiki/Timelike" class="mw-redirect" title="Timelike">timelike</a> <a href="/wiki/World_line" title="World line">world line</a> is defined as the <a href="/wiki/Time" title="Time">time</a> as measured by a <a href="/wiki/Clock" title="Clock">clock</a> following that line. The <b>proper time interval</b> between two <a href="/wiki/Event_(relativity)" title="Event (relativity)">events</a> on a world line is the change in proper time, which is independent of coordinates, and is a <a href="/wiki/Lorentz_scalar" title="Lorentz scalar">Lorentz scalar</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line. </p><p>The proper time interval between two events depends not only on the events, but also the world line connecting them, and hence on the motion of the clock between the events. It is expressed as an integral over the world line (analogous to <a href="/wiki/Arc_length" title="Arc length">arc length</a> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>). An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (<a href="/wiki/Inertial" class="mw-redirect" title="Inertial">inertial</a>) clock between the same two events. The <a href="/wiki/Twin_paradox" title="Twin paradox">twin paradox</a> is an example of this effect.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Proper_and_coordinate_time.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Proper_and_coordinate_time.png/220px-Proper_and_coordinate_time.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Proper_and_coordinate_time.png/330px-Proper_and_coordinate_time.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Proper_and_coordinate_time.png/440px-Proper_and_coordinate_time.png 2x" data-file-width="556" data-file-height="579" /></a><figcaption>The dark blue vertical line represents an inertial observer measuring a coordinate time interval <i>t</i> between events <i>E</i><sub>1</sub> and <i>E</i><sub>2</sub>. The red curve represents a clock measuring its proper time interval <i>τ</i> between the same two events.</figcaption></figure> <p>By convention, proper time is usually represented by the Greek letter <i>τ</i> (<a href="/wiki/Tau" title="Tau">tau</a>) to distinguish it from <a href="/wiki/Coordinate_time" title="Coordinate time">coordinate time</a> represented by <i>t</i>. Coordinate time is the time between two events as measured by an observer using that observer's own method of assigning a time to an event. In the special case of an inertial observer in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>, the time is measured using the observer's clock and the observer's definition of simultaneity. </p><p>The concept of proper time was introduced by <a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Hermann Minkowski</a> in 1908,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> and is an important feature of <a href="/wiki/Minkowski_diagram" class="mw-redirect" title="Minkowski diagram">Minkowski diagrams</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Mathematical_formalism">Mathematical formalism</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=1" title="Edit section: Mathematical formalism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formal definition of proper time involves describing the path through <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> that represents a clock, observer, or test particle, and the <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric structure</a> of that spacetime. Proper time is the <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian</a> arc length of <a href="/wiki/World_line" title="World line">world lines</a> in four-dimensional spacetime. From the mathematical point of view, coordinate time is assumed to be predefined and an expression for proper time as a function of coordinate time is required. On the other hand, proper time is measured experimentally and coordinate time is calculated from the proper time of inertial clocks. </p><p>Proper time can only be defined for timelike paths through spacetime which allow for the construction of an accompanying set of physical rulers and clocks. The same formalism for spacelike paths leads to a measurement of <a href="/wiki/Proper_distance" class="mw-redirect" title="Proper distance">proper distance</a> rather than proper time. For lightlike paths, there exists no concept of proper time and it is undefined as the spacetime interval is zero. Instead, an arbitrary and physically irrelevant <a href="/wiki/Geodesics" class="mw-redirect" title="Geodesics">affine parameter</a> unrelated to time must be introduced.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_special_relativity">In special relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=2" title="Edit section: In special relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>With the <a href="/wiki/Timelike" class="mw-redirect" title="Timelike">timelike</a> convention for the <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a>, the <a href="/wiki/Minkowski_metric" class="mw-redirect" title="Minkowski metric">Minkowski metric</a> is defined by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d64df60a94641aba237aba2a34b024cd64548da8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:28.856ex; height:12.509ex;" alt="{\displaystyle \eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}},}"></span> and the coordinates by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f4943384dd2ee423b6b60b4c9114fa8075213bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.876ex; height:3.176ex;" alt="{\displaystyle (x^{0},x^{1},x^{2},x^{3})=(ct,x,y,z)}"></span> for arbitrary Lorentz frames. </p><p>In any such frame an infinitesimal interval, here assumed timelike, between two events is expressed as </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk numblk-raw-n" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32123bea1c4ba1628067bec1156c56982d6e77f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.955ex; height:3.343ex;" alt="{\displaystyle ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu },}"></span> </td> <td></td> <td class="nowrap"><span id="math_(1)" class="reference nourlexpansion" style="font-weight:bold;">(1)</span></td></tr></tbody></table> <p>and separates points on a trajectory of a particle (think clock{?}). The same interval can be expressed in coordinates such that at each moment, the particle is <i>at rest</i>. Such a frame is called an instantaneous rest frame, denoted here by the coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c\tau ,x_{\tau },y_{\tau },z_{\tau })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>c</mi> <mi>τ</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c\tau ,x_{\tau },y_{\tau },z_{\tau })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0642aeb035134479ae1f3553832f783064ac74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.916ex; height:2.843ex;" alt="{\displaystyle (c\tau ,x_{\tau },y_{\tau },z_{\tau })}"></span> for each instant. Due to the invariance of the interval (instantaneous rest frames taken at different times are related by Lorentz transformations) one may write <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}=c^{2}d\tau ^{2}-dx_{\tau }^{2}-dy_{\tau }^{2}-dz_{\tau }^{2}=c^{2}d\tau ^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>d</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>d</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mi>d</mi> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}=c^{2}d\tau ^{2}-dx_{\tau }^{2}-dy_{\tau }^{2}-dz_{\tau }^{2}=c^{2}d\tau ^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f50d0739300346fa2e9e700833b4029f0b8305be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.348ex; height:3.009ex;" alt="{\displaystyle ds^{2}=c^{2}d\tau ^{2}-dx_{\tau }^{2}-dy_{\tau }^{2}-dz_{\tau }^{2}=c^{2}d\tau ^{2},}"></span> since in the instantaneous rest frame, the particle or the frame itself is at rest, i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx_{\tau }=dy_{\tau }=dz_{\tau }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <msub> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx_{\tau }=dy_{\tau }=dz_{\tau }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd4e06c8a75d6cf684c9dc88137c87d8f63529b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.902ex; height:2.509ex;" alt="{\displaystyle dx_{\tau }=dy_{\tau }=dz_{\tau }=0}"></span>. Since the interval is assumed timelike (ie. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds^{2}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds^{2}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7557284c7e1fe19d22dbb5eda021ca551c432ac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.621ex; height:2.676ex;" alt="{\displaystyle ds^{2}>0}"></span>), taking the square root of the above yields<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ds=cd\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>s</mi> <mo>=</mo> <mi>c</mi> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ds=cd\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fc97114a03a8f1cea104d60ae15df49fea0002d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.476ex; height:2.509ex;" alt="{\displaystyle ds=cd\tau ,}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\frac {ds}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\frac {ds}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51187269e926a2c96b4cd3ae06d22ab34e7a085e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.306ex; height:5.343ex;" alt="{\displaystyle d\tau ={\frac {ds}{c}}.}"></span> Given this differential expression for <span class="texhtml mvar" style="font-style:italic;">τ</span>, the proper time interval is defined as </p> <div class="equation-box" style="margin: 0;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {ds}{c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>s</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {ds}{c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a728bce401117ddc3b0857dbfabc9ead93f4768" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.834ex; height:5.843ex;" alt="{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {ds}{c}}.}"></span><span class="nowrap">          </span><span id="math_(2)" class="reference nourlexpansion" style="font-weight:bold;">(2)</span> </p> </div> <p>Here <span class="texhtml mvar" style="font-style:italic;">P</span> is the worldline from some initial event to some final event with the ordering of the events fixed by the requirement that the final event occurs later according to the clock than the initial event. </p><p>Using <b><a href="#math_(1)">(1)</a></b> and again the invariance of the interval, one may write<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="equation-box" style="margin: 0;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\Delta \tau &=\int _{P}{\frac {1}{c}}{\sqrt {\eta _{\mu \nu }dx^{\mu }dx^{\nu }}}\\&=\int _{P}{\sqrt {dt^{2}-{dx^{2} \over c^{2}}-{dy^{2} \over c^{2}}-{dz^{2} \over c^{2}}}}\\&=\int _{a}^{b}{\sqrt {1-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\right]}}dt\\&=\int _{a}^{b}{\sqrt {1-{\frac {v(t)^{2}}{c^{2}}}}}dt\\&=\int _{a}^{b}{\frac {dt}{\gamma (t)}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </msqrt> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Delta \tau &=\int _{P}{\frac {1}{c}}{\sqrt {\eta _{\mu \nu }dx^{\mu }dx^{\nu }}}\\&=\int _{P}{\sqrt {dt^{2}-{dx^{2} \over c^{2}}-{dy^{2} \over c^{2}}-{dz^{2} \over c^{2}}}}\\&=\int _{a}^{b}{\sqrt {1-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\right]}}dt\\&=\int _{a}^{b}{\sqrt {1-{\frac {v(t)^{2}}{c^{2}}}}}dt\\&=\int _{a}^{b}{\frac {dt}{\gamma (t)}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11f62e17762411e113c9452ef6e343ef5d158f7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.671ex; width:54.12ex; height:36.509ex;" alt="{\displaystyle {\begin{aligned}\Delta \tau &=\int _{P}{\frac {1}{c}}{\sqrt {\eta _{\mu \nu }dx^{\mu }dx^{\nu }}}\\&=\int _{P}{\sqrt {dt^{2}-{dx^{2} \over c^{2}}-{dy^{2} \over c^{2}}-{dz^{2} \over c^{2}}}}\\&=\int _{a}^{b}{\sqrt {1-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}\right]}}dt\\&=\int _{a}^{b}{\sqrt {1-{\frac {v(t)^{2}}{c^{2}}}}}dt\\&=\int _{a}^{b}{\frac {dt}{\gamma (t)}},\end{aligned}}}"></span><span class="nowrap">          </span><span id="math_(3)" class="reference nourlexpansion" style="font-weight:bold;">(3)</span> </p> </div> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{0},x^{1},x^{2},x^{3}):[a,b]\rightarrow P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{0},x^{1},x^{2},x^{3}):[a,b]\rightarrow P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f1ad38fffdf7efb27d3e69a9724e36643c2a15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.298ex; height:3.176ex;" alt="{\displaystyle (x^{0},x^{1},x^{2},x^{3}):[a,b]\rightarrow P}"></span> is an arbitrary bijective parametrization of the worldline <span class="texhtml mvar" style="font-style:italic;">P</span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{0}(a),x^{1}(a),x^{2}(a),x^{3}(a))\quad {\text{and}}\quad (x^{0}(b),x^{1}(b),x^{2}(b),x^{3}(b))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{0}(a),x^{1}(a),x^{2}(a),x^{3}(a))\quad {\text{and}}\quad (x^{0}(b),x^{1}(b),x^{2}(b),x^{3}(b))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd05ea662799c7a1797ee66d4e384b0a72412ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.67ex; height:3.176ex;" alt="{\displaystyle (x^{0}(a),x^{1}(a),x^{2}(a),x^{3}(a))\quad {\text{and}}\quad (x^{0}(b),x^{1}(b),x^{2}(b),x^{3}(b))}"></span> give the endpoints of <span class="texhtml mvar" style="font-style:italic;">P</span> and a < b; <span class="texhtml"><i>v</i>(<i>t</i>)</span> is the coordinate speed at coordinate time <span class="texhtml mvar" style="font-style:italic;">t</span>; and <span class="texhtml"><i>x</i>(<i>t</i>)</span>, <span class="texhtml"><i>y</i>(<i>t</i>)</span>, and <span class="texhtml"><i>z</i>(<i>t</i>)</span> are space coordinates. The first expression is <i>manifestly</i> Lorentz invariant. They are all Lorentz invariant, since proper time and proper time intervals are coordinate-independent by definition. </p><p>If <span class="texhtml"><i>t</i>, <i>x</i>, <i>y</i>, <i>z</i></span>, are parameterised by a <a href="/wiki/Parameter" title="Parameter">parameter</a> <span class="texhtml mvar" style="font-style:italic;">λ</span>, this can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau =\int {\sqrt {\left({\frac {dt}{d\lambda }}\right)^{2}-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{d\lambda }}\right)^{2}+\left({\frac {dy}{d\lambda }}\right)^{2}+\left({\frac {dz}{d\lambda }}\right)^{2}\right]}}\,d\lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau =\int {\sqrt {\left({\frac {dt}{d\lambda }}\right)^{2}-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{d\lambda }}\right)^{2}+\left({\frac {dy}{d\lambda }}\right)^{2}+\left({\frac {dz}{d\lambda }}\right)^{2}\right]}}\,d\lambda .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c40f6c7ed66949d90f684ec4bd8e2ed8042ef5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:60.922ex; height:8.176ex;" alt="{\displaystyle \Delta \tau =\int {\sqrt {\left({\frac {dt}{d\lambda }}\right)^{2}-{\frac {1}{c^{2}}}\left[\left({\frac {dx}{d\lambda }}\right)^{2}+\left({\frac {dy}{d\lambda }}\right)^{2}+\left({\frac {dz}{d\lambda }}\right)^{2}\right]}}\,d\lambda .}"></span> </p><p>If the motion of the particle is constant, the expression simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau ={\sqrt {\left(\Delta t\right)^{2}-{\frac {\left(\Delta x\right)^{2}}{c^{2}}}-{\frac {\left(\Delta y\right)^{2}}{c^{2}}}-{\frac {\left(\Delta z\right)^{2}}{c^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">Δ</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau ={\sqrt {\left(\Delta t\right)^{2}-{\frac {\left(\Delta x\right)^{2}}{c^{2}}}-{\frac {\left(\Delta y\right)^{2}}{c^{2}}}-{\frac {\left(\Delta z\right)^{2}}{c^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209b7c07e25237dd126124d861fd69c9b2ab39d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.847ex; height:7.509ex;" alt="{\displaystyle \Delta \tau ={\sqrt {\left(\Delta t\right)^{2}-{\frac {\left(\Delta x\right)^{2}}{c^{2}}}-{\frac {\left(\Delta y\right)^{2}}{c^{2}}}-{\frac {\left(\Delta z\right)^{2}}{c^{2}}}}},}"></span> where Δ means the change in coordinates between the initial and final events. The definition in special relativity generalizes straightforwardly to general relativity as follows below. </p> <div class="mw-heading mw-heading3"><h3 id="In_general_relativity">In general relativity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=3" title="Edit section: In general relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Proper time is defined in <a href="/wiki/General_relativity" title="General relativity">general relativity</a> as follows: Given a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a> with a local coordinates <span class="texhtml"><i>x</i><sup><i>μ</i></sup></span> and equipped with a <a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">metric tensor</a> <span class="texhtml"><i>g</i><sub><i>μν</i></sub></span>, the proper time interval <span class="texhtml">Δ<i>τ</i></span> between two events along a timelike path <span class="texhtml mvar" style="font-style:italic;">P</span> is given by the <a href="/wiki/Line_integral" title="Line integral">line integral</a><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk numblk-raw-n" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau =\int _{P}\,d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{\mu \nu }\;dx^{\mu }\;dx^{\nu }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau =\int _{P}\,d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{\mu \nu }\;dx^{\mu }\;dx^{\nu }}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e17d1bb2f90e0d9b97891149bd72f18ab2ebdbd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:35.313ex; height:5.676ex;" alt="{\displaystyle \Delta \tau =\int _{P}\,d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{\mu \nu }\;dx^{\mu }\;dx^{\nu }}}.}"></span></td> <td></td> <td class="nowrap"><span id="math_(4)" class="reference nourlexpansion" style="font-weight:bold;">(4)</span></td></tr></tbody></table> <p>This expression is, as it should be, invariant under coordinate changes. It reduces (in appropriate coordinates) to the expression of special relativity in <a href="/wiki/Flat_spacetime" class="mw-redirect" title="Flat spacetime">flat spacetime</a>. </p><p>In the same way that coordinates can be chosen such that <span class="texhtml"><i>x</i><sup>1</sup>, <i>x</i><sup>2</sup>, <i>x</i><sup>3</sup> = const</span> in special relativity, this can be done in general relativity too. Then, in these coordinates,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{00}}}dx^{0}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>00</mn> </mrow> </msub> </msqrt> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{00}}}dx^{0}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec172b449e4256bdba60010fef066c09303eb782" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.211ex; height:5.676ex;" alt="{\displaystyle \Delta \tau =\int _{P}d\tau =\int _{P}{\frac {1}{c}}{\sqrt {g_{00}}}dx^{0}.}"></span> </p><p>This expression generalizes definition <b><a href="#math_(2)">(2)</a></b> and can be taken as the definition. Then using invariance of the interval, equation <b><a href="#math_(4)">(4)</a></b> follows from it in the same way <b><a href="#math_(3)">(3)</a></b> follows from <b><a href="#math_(2)">(2)</a></b>, except that here arbitrary coordinate changes are allowed. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_in_special_relativity">Examples in special relativity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=4" title="Edit section: Examples in special relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Example_1:_The_twin_"paradox""><span id="Example_1:_The_twin_.22paradox.22"></span>Example 1: The twin "paradox"</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=5" title="Edit section: Example 1: The twin "paradox""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a <a href="/wiki/Twin_paradox" title="Twin paradox">twin paradox</a> scenario, let there be an observer <i>A</i> who moves between the <i>A</i>-coordinates (0,0,0,0) and (10 years, 0, 0, 0) inertially. This means that <i>A</i> stays at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y=z=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y=z=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806816016dd7dfb90cb1cd0560506c14a29e2535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.031ex; height:2.509ex;" alt="{\displaystyle x=y=z=0}"></span> for 10 years of <i>A</i>-coordinate time. The proper time interval for <i>A</i> between the two events is then <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau _{A}={\sqrt {(10{\text{ years}})^{2}}}=10{\text{ years}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> years</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> years</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau _{A}={\sqrt {(10{\text{ years}})^{2}}}=10{\text{ years}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59953b2dc75c6ea40399208d84537d95aa02c71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:32.759ex; height:4.843ex;" alt="{\displaystyle \Delta \tau _{A}={\sqrt {(10{\text{ years}})^{2}}}=10{\text{ years}}.}"></span> </p><p>So being "at rest" in a special relativity coordinate system means that proper time and coordinate time are the same. </p><p>Let there now be another observer <i>B</i> who travels in the <i>x</i> direction from (0,0,0,0) for 5 years of <i>A</i>-coordinate time at 0.866<i>c</i> to (5 years, 4.33 light-years, 0, 0). Once there, <i>B</i> accelerates, and travels in the other spatial direction for another 5 years of <i>A</i>-coordinate time to (10 years, 0, 0, 0). For each leg of the trip, the proper time interval can be calculated using <i>A</i>-coordinates, and is given by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau _{leg}={\sqrt {({\text{5 years}})^{2}-({\text{4.33 years}})^{2}}}={\sqrt {6.25\;\mathrm {years} ^{2}}}={\text{2.5 years}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>e</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>5 years</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>4.33 years</mtext> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>6.25</mn> <mspace width="thickmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>2.5 years</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau _{leg}={\sqrt {({\text{5 years}})^{2}-({\text{4.33 years}})^{2}}}={\sqrt {6.25\;\mathrm {years} ^{2}}}={\text{2.5 years}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad389a422203f798eb341bb25633cc2e5f372739" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:65.232ex; height:4.843ex;" alt="{\displaystyle \Delta \tau _{leg}={\sqrt {({\text{5 years}})^{2}-({\text{4.33 years}})^{2}}}={\sqrt {6.25\;\mathrm {years} ^{2}}}={\text{2.5 years}}.}"></span> </p><p>So the total proper time for observer <i>B</i> to go from (0,0,0,0) to (5 years, 4.33 light-years, 0, 0) and then to (10 years, 0, 0, 0) is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau _{B}=2\Delta \tau _{leg}={\text{5 years}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi mathvariant="normal">Δ</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>e</mi> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>5 years</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau _{B}=2\Delta \tau _{leg}={\text{5 years}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3916b5aafd0d27afaa191c5ace1789a76770710" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.661ex; height:2.843ex;" alt="{\displaystyle \Delta \tau _{B}=2\Delta \tau _{leg}={\text{5 years}}.}"></span> </p><p>Thus it is shown that the proper time equation incorporates the <a href="/wiki/Time_dilation" title="Time dilation">time dilation</a> effect. In fact, for an object in a SR (special relativity) spacetime traveling with velocity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> for a time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e61e7deb9c7c7b7dda762b0935e757add2acc559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.572ex; height:2.176ex;" alt="{\displaystyle \Delta T}"></span>, the proper time interval experienced is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \tau ={\sqrt {\Delta T^{2}-\left({\frac {v_{x}\Delta T}{c}}\right)^{2}-\left({\frac {v_{y}\Delta T}{c}}\right)^{2}-\left({\frac {v_{z}\Delta T}{c}}\right)^{2}}}=\Delta T{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">Δ</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mi mathvariant="normal">Δ</mi> <mi>T</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \tau ={\sqrt {\Delta T^{2}-\left({\frac {v_{x}\Delta T}{c}}\right)^{2}-\left({\frac {v_{y}\Delta T}{c}}\right)^{2}-\left({\frac {v_{z}\Delta T}{c}}\right)^{2}}}=\Delta T{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cff08bc13c97413738a954acaa922abd96ecbee6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.711ex; height:7.676ex;" alt="{\displaystyle \Delta \tau ={\sqrt {\Delta T^{2}-\left({\frac {v_{x}\Delta T}{c}}\right)^{2}-\left({\frac {v_{y}\Delta T}{c}}\right)^{2}-\left({\frac {v_{z}\Delta T}{c}}\right)^{2}}}=\Delta T{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},}"></span> which is the SR time dilation formula. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2:_The_rotating_disk">Example 2: The rotating disk</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=6" title="Edit section: Example 2: The rotating disk"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An observer rotating around another inertial observer is in an accelerated frame of reference. For such an observer, the incremental (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00568785317ea373b90759c05c67d795b57b3194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:2.176ex;" alt="{\displaystyle d\tau }"></span>) form of the proper time equation is needed, along with a parameterized description of the path being taken, as shown below. </p><p>Let there be an observer <i>C</i> on a disk rotating in the <i>xy</i> plane at a coordinate angular rate of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> and who is at a distance of <i>r</i> from the center of the disk with the center of the disk at <span class="texhtml"><i>x</i> = <i>y</i> = <i>z</i> = 0</span>. The path of observer <i>C</i> is given by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,\,r\cos(\omega T),\,r\sin(\omega T),\,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>r</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>r</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T,\,r\cos(\omega T),\,r\sin(\omega T),\,0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eec7b44f33f2434af14670cfaa4ed7073ecef704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.492ex; height:2.843ex;" alt="{\displaystyle (T,\,r\cos(\omega T),\,r\sin(\omega T),\,0)}"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is the current coordinate time. When <i>r</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> are constant, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx=-r\omega \sin(\omega T)\,dT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>r</mi> <mi>ω</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx=-r\omega \sin(\omega T)\,dT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7908b79e20431ca53aee0a4c8d3ffe8a65d4524b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.32ex; height:2.843ex;" alt="{\displaystyle dx=-r\omega \sin(\omega T)\,dT}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dy=r\omega \cos(\omega T)\,dT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mi>r</mi> <mi>ω</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dy=r\omega \cos(\omega T)\,dT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/329074934455368cfed07ad4c98b180f9c6d6cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.593ex; height:2.843ex;" alt="{\displaystyle dy=r\omega \cos(\omega T)\,dT}"></span>. The incremental proper time formula then becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\sin ^{2}(\omega T)\;dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\cos ^{2}(\omega T)\;dT^{2}}}=dT{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>cos</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mi>d</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>d</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\sin ^{2}(\omega T)\;dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\cos ^{2}(\omega T)\;dT^{2}}}=dT{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed549ba9c74f748cf5c9de280d5a3254a6ec88d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:79.819ex; height:6.176ex;" alt="{\displaystyle d\tau ={\sqrt {dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\sin ^{2}(\omega T)\;dT^{2}-\left({\frac {r\omega }{c}}\right)^{2}\cos ^{2}(\omega T)\;dT^{2}}}=dT{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}.}"></span> </p><p>So for an observer rotating at a constant distance of <i>r</i> from a given point in spacetime at a constant angular rate of <i>ω</i> between coordinate times <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f304724948a3ef606c4a92459e22b87a954d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1ba5f12fbb0ff766aec6e22148b429373608555" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{2}}"></span>, the proper time experienced will be <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{T_{1}}^{T_{2}}d\tau =(T_{2}-T_{1}){\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}=\Delta T{\sqrt {1-v^{2}/c^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{T_{1}}^{T_{2}}d\tau =(T_{2}-T_{1}){\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}=\Delta T{\sqrt {1-v^{2}/c^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2822910e0679b06451613fa49d94211569ac5521" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.401ex; height:6.509ex;" alt="{\displaystyle \int _{T_{1}}^{T_{2}}d\tau =(T_{2}-T_{1}){\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}}=\Delta T{\sqrt {1-v^{2}/c^{2}}},}"></span> as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v=r\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mi>r</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v=r\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b5b5c066f5be8dee84c4cb33dea2383de8b012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:1.676ex;" alt="{\displaystyle v=r\omega }"></span> for a rotating observer. This result is the same as for the linear motion example, and shows the general application of the integral form of the proper time formula. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_in_general_relativity">Examples in general relativity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=7" title="Edit section: Examples in general relativity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The difference between SR and general relativity (GR) is that in GR one can use any metric which is a solution of the <a href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</a>, not just the Minkowski metric. Because inertial motion in curved spacetimes lacks the simple expression it has in SR, the line integral form of the proper time equation must always be used. </p> <div class="mw-heading mw-heading3"><h3 id="Example_3:_The_rotating_disk_(again)"><span id="Example_3:_The_rotating_disk_.28again.29"></span>Example 3: The rotating disk (again)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=8" title="Edit section: Example 3: The rotating disk (again)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An appropriate <a href="/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates" title="Polar coordinate system">coordinate conversion</a> done against the Minkowski metric creates coordinates where an object on a rotating disk stays in the same spatial coordinate position. The new coordinates are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\sqrt {x^{2}+y^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\sqrt {x^{2}+y^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66fcac81cfac010069078ce8c999bd09f285567f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.91ex; height:4.843ex;" alt="{\displaystyle r={\sqrt {x^{2}+y^{2}}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\arctan \left({\frac {y}{x}}\right)-\omega t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>y</mi> <mi>x</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\arctan \left({\frac {y}{x}}\right)-\omega t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b32277f3b700d510d22356b141613696d288526f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.369ex; height:4.843ex;" alt="{\displaystyle \theta =\arctan \left({\frac {y}{x}}\right)-\omega t.}"></span> </p><p>The <i>t</i> and <i>z</i> coordinates remain unchanged. In this new coordinate system, the incremental proper time equation is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {r\omega }{c}}\right)^{2}\right]dt^{2}-{\frac {dr^{2}}{c^{2}}}-{\frac {r^{2}\,d\theta ^{2}}{c^{2}}}-{\frac {dz^{2}}{c^{2}}}-2{\frac {r^{2}\omega \,dt\,d\theta }{c^{2}}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>θ</mi> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {r\omega }{c}}\right)^{2}\right]dt^{2}-{\frac {dr^{2}}{c^{2}}}-{\frac {r^{2}\,d\theta ^{2}}{c^{2}}}-{\frac {dz^{2}}{c^{2}}}-2{\frac {r^{2}\omega \,dt\,d\theta }{c^{2}}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbd824f14c0bdcd7c65a07a7b65328d116c8689" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.685ex; height:7.509ex;" alt="{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {r\omega }{c}}\right)^{2}\right]dt^{2}-{\frac {dr^{2}}{c^{2}}}-{\frac {r^{2}\,d\theta ^{2}}{c^{2}}}-{\frac {dz^{2}}{c^{2}}}-2{\frac {r^{2}\omega \,dt\,d\theta }{c^{2}}}}}.}"></span> </p><p>With <i>r</i>, <i>θ</i>, and <i>z</i> being constant over time, this simplifies to <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau =dt{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mi>d</mi> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau =dt{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1884cb6972f7bffce5cd3bb9563c1c0133ca9cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.706ex; height:6.176ex;" alt="{\displaystyle d\tau =dt{\sqrt {1-\left({\frac {r\omega }{c}}\right)^{2}}},}"></span> which is the same as in Example 2. </p><p>Now let there be an object off of the rotating disk and at inertial rest with respect to the center of the disk and at a distance of <i>R</i> from it. This object has a <b>coordinate</b> motion described by <span class="texhtml"><i>dθ</i> = −<i>ω</i> <i>dt</i></span>, which describes the inertially at-rest object of counter-rotating in the view of the rotating observer. Now the proper time equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {R\omega }{c}}\right)^{2}\right]dt^{2}-\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}+2\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}}}=dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>ω</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {R\omega }{c}}\right)^{2}\right]dt^{2}-\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}+2\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}}}=dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa6583b13535277bdf70cae2f307839bbdd81506" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.382ex; height:8.176ex;" alt="{\displaystyle d\tau ={\sqrt {\left[1-\left({\frac {R\omega }{c}}\right)^{2}\right]dt^{2}-\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}+2\left({\frac {R\omega }{c}}\right)^{2}\,dt^{2}}}=dt.}"></span> </p><p>So for the inertial at-rest observer, coordinate time and proper time are once again found to pass at the same rate, as expected and required for the internal self-consistency of relativity theory.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Example_4:_The_Schwarzschild_solution_–_time_on_the_Earth"><span id="Example_4:_The_Schwarzschild_solution_.E2.80.93_time_on_the_Earth"></span>Example 4: The Schwarzschild solution – time on the Earth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=9" title="Edit section: Example 4: The Schwarzschild solution – time on the Earth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Schwarzschild_solution" class="mw-redirect" title="Schwarzschild solution">Schwarzschild solution</a> has an incremental proper time equation of <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {\left(1-{\frac {2m}{r}}\right)dt^{2}-{\frac {1}{c^{2}}}\left(1-{\frac {2m}{r}}\right)^{-1}dr^{2}-{\frac {r^{2}}{c^{2}}}d\phi ^{2}-{\frac {r^{2}}{c^{2}}}\sin ^{2}(\phi )\,d\theta ^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>m</mi> </mrow> <mi>r</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>d</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mi>d</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {\left(1-{\frac {2m}{r}}\right)dt^{2}-{\frac {1}{c^{2}}}\left(1-{\frac {2m}{r}}\right)^{-1}dr^{2}-{\frac {r^{2}}{c^{2}}}d\phi ^{2}-{\frac {r^{2}}{c^{2}}}\sin ^{2}(\phi )\,d\theta ^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773cd2f47d80598614cc5a9298fc44ea543061c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:72.753ex; height:7.676ex;" alt="{\displaystyle d\tau ={\sqrt {\left(1-{\frac {2m}{r}}\right)dt^{2}-{\frac {1}{c^{2}}}\left(1-{\frac {2m}{r}}\right)^{-1}dr^{2}-{\frac {r^{2}}{c^{2}}}d\phi ^{2}-{\frac {r^{2}}{c^{2}}}\sin ^{2}(\phi )\,d\theta ^{2}}},}"></span> where </p> <ul><li><i>t</i> is time as calibrated with a clock distant from and at inertial rest with respect to the Earth,</li> <li><i>r</i> is a radial coordinate (which is effectively the distance from the Earth's center),</li> <li><i>ɸ</i> is a co-latitudinal coordinate, the angular separation from the <a href="/wiki/North_pole" class="mw-redirect" title="North pole">north pole</a> in <a href="/wiki/Radian" title="Radian">radians</a>.</li> <li><i>θ</i> is a longitudinal coordinate, analogous to the longitude on the Earth's surface but independent of the Earth's <a href="/wiki/Rotation" title="Rotation">rotation</a>. This is also given in radians.</li> <li><i>m</i> is the <a href="/wiki/Geometrized" class="mw-redirect" title="Geometrized">geometrized</a> mass of the Earth, <i>m</i> = <i>GM</i>/<i>c</i><sup>2</sup>, <ul><li><i>M</i> is the mass of the Earth,</li> <li><i>G</i> is the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a>.</li></ul></li></ul> <p>To demonstrate the use of the proper time relationship, several sub-examples involving the Earth will be used here. </p><p>For the <a href="/wiki/Earth" title="Earth">Earth</a>, <span class="texhtml"><i>M</i> = <span class="nowrap"><span data-sort-value="7024597420000000000♠"></span>5.9742<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>24</sup> kg</span></span>, meaning that <span class="texhtml"><i>m</i> = <span class="nowrap"><span data-sort-value="6997443540000000000♠"></span>4.4354<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−3</sup> m</span></span>. When standing on the north pole, we can assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dr=d\theta =d\phi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>r</mi> <mo>=</mo> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mi>d</mi> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dr=d\theta =d\phi =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7db64d02b0f9c8b075690d0f9189ee1c8d39574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.63ex; height:2.509ex;" alt="{\displaystyle dr=d\theta =d\phi =0}"></span> (meaning that we are neither moving up or down or along the surface of the Earth). In this case, the Schwarzschild solution proper time equation becomes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d\tau =dt\,{\sqrt {1-2m/r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mi>d</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d\tau =dt\,{\sqrt {1-2m/r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480919ee64be7373f8c1565f63313debff8988cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.699ex; height:3.343ex;" alt="{\textstyle d\tau =dt\,{\sqrt {1-2m/r}}}"></span>. Then using the polar radius of the Earth as the radial coordinate (or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r={\text{6,356,752 metres}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>6,356,752 metres</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r={\text{6,356,752 metres}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4e77f298e12e922ed75cb111a49fd5397cc57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.892ex; height:2.509ex;" alt="{\displaystyle r={\text{6,356,752 metres}}}"></span>), we find that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)\;dt^{2}}}=\left(1-6.9540\times 10^{-10}\right)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1.3908</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>6.9540</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)\;dt^{2}}}=\left(1-6.9540\times 10^{-10}\right)\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2722652651aac39d4086700d730da3c7837dd06c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:59.559ex; height:4.843ex;" alt="{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)\;dt^{2}}}=\left(1-6.9540\times 10^{-10}\right)\,dt.}"></span> </p><p>At the <a href="/wiki/Equator" title="Equator">equator</a>, the radius of the Earth is <span class="texhtml"><i>r</i> = <span class="nowrap"><span data-sort-value="7006637813700000000♠"></span>6<span style="margin-left:.25em;">378</span><span style="margin-left:.25em;">137</span> m</span></span>. In addition, the rotation of the Earth needs to be taken into account. This imparts on an observer an angular velocity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta /dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta /dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe083af9808068de003c10018962791d930c010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.524ex; height:2.843ex;" alt="{\displaystyle d\theta /dt}"></span> of 2<i>π</i> divided by the <a href="/wiki/Sidereal_time" title="Sidereal time">sidereal period</a> of the Earth's rotation, 86162.4 seconds. So <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\theta =7.2923\times 10^{-5}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>θ</mi> <mo>=</mo> <mn>7.2923</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\theta =7.2923\times 10^{-5}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbd63ad4ea26529bd363494b0016cf92f5ae8448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.805ex; height:2.676ex;" alt="{\displaystyle d\theta =7.2923\times 10^{-5}\,dt}"></span>. The proper time equation then produces <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)dt^{2}-2.4069\times 10^{-12}\,dt^{2}}}=\left(1-6.9660\times 10^{-10}\right)\,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>1.3908</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2.4069</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>12</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>6.9660</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>10</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)dt^{2}-2.4069\times 10^{-12}\,dt^{2}}}=\left(1-6.9660\times 10^{-10}\right)\,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a27e03a3953404f4fdd3415efd0e14caccf436" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:80.03ex; height:4.843ex;" alt="{\displaystyle d\tau ={\sqrt {\left(1-1.3908\times 10^{-9}\right)dt^{2}-2.4069\times 10^{-12}\,dt^{2}}}=\left(1-6.9660\times 10^{-10}\right)\,dt.}"></span> </p><p>From a non-relativistic point of view this should have been the same as the previous result. This example demonstrates how the proper time equation is used, even though the Earth rotates and hence is not spherically symmetric as assumed by the Schwarzschild solution. To describe the effects of rotation more accurately the <a href="/wiki/Kerr_metric" title="Kerr metric">Kerr metric</a> may be used. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a></li> <li><a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a></li> <li><a href="/wiki/Proper_length" title="Proper length">Proper length</a></li> <li><a href="/wiki/Proper_acceleration" title="Proper acceleration">Proper acceleration</a></li> <li><a href="/wiki/Invariant_mass" title="Invariant mass">Proper mass</a></li> <li><a href="/wiki/Proper_velocity" title="Proper velocity">Proper velocity</a></li> <li><a href="/wiki/Clock_hypothesis" class="mw-redirect" title="Clock hypothesis">Clock hypothesis</a></li> <li><a href="/wiki/Peres_metric" title="Peres metric">Peres metric</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=11" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFZwiebach2004">Zwiebach 2004</a>, p. 25</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHawleyHolcomb2005" class="citation book cs1">Hawley, John F.; Holcomb, J Katherine A. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s5MUDAAAQBAJ"><i>Foundations of Modern Cosmology</i></a> (illustrated ed.). Oxford University Press. p. 204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853096-1" title="Special:BookSources/978-0-19-853096-1"><bdi>978-0-19-853096-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Modern+Cosmology&rft.pages=204&rft.edition=illustrated&rft.pub=Oxford+University+Press&rft.date=2005&rft.isbn=978-0-19-853096-1&rft.aulast=Hawley&rft.aufirst=John+F.&rft.au=Holcomb%2C+J+Katherine+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ds5MUDAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=s5MUDAAAQBAJ&pg=PA204">Extract of page 204</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFMinkowski1908">Minkowski 1908</a>, pp. 53–111</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFLovelockRund1989">Lovelock & Rund 1989</a>, pp. 256</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFWeinberg1972">Weinberg 1972</a>, pp. 76</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFPoisson2004">Poisson 2004</a>, pp. 7</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFLandauLifshitz1975">Landau & Lifshitz 1975</a>, p. 245</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Some authors include lightlike intervals in the definition of proper time, and also include the spacelike proper distances as imaginary proper times e.g <a href="#CITEREFLawden2012">Lawden 2012</a>, pp. 17, 116</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><a href="#CITEREFKopeikinEfroimskyKaplan2011">Kopeikin, Efroimsky & Kaplan 2011</a>, p. 275</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFZwiebach2004">Zwiebach 2004</a>, p. 25</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFFosterNightingale1978">Foster & Nightingale 1978</a>, p. 56</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFFosterNightingale1978">Foster & Nightingale 1978</a>, p. 57</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFLandauLifshitz1975">Landau & Lifshitz 1975</a>, p. 251</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFCook2004">Cook 2004</a>, pp. 214–219</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Proper_time&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCook2004" class="citation journal cs1">Cook, R. J. (2004). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1236110">"Physical time and physical space in general relativity"</a>. <i>Am. J. Phys</i>. <b>72</b> (2): <span class="nowrap">214–</span>219. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004AmJPh..72..214C">2004AmJPh..72..214C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1607338">10.1119/1.1607338</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9505">0002-9505</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am.+J.+Phys.&rft.atitle=Physical+time+and+physical+space+in+general+relativity&rft.volume=72&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E214-%3C%2Fspan%3E219&rft.date=2004&rft.issn=0002-9505&rft_id=info%3Adoi%2F10.1119%2F1.1607338&rft_id=info%3Abibcode%2F2004AmJPh..72..214C&rft.aulast=Cook&rft.aufirst=R.+J.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1236110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFosterNightingale1978" class="citation book cs1">Foster, J.; Nightingale, J.D. (1978). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/shortcourseingen0000fost"><i>A short course in general relativity</i></a></span>. Essex: <a href="/wiki/Longman_Scientific_and_Technical" class="mw-redirect" title="Longman Scientific and Technical">Longman Scientific and Technical</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-44194-3" title="Special:BookSources/0-582-44194-3"><bdi>0-582-44194-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+short+course+in+general+relativity&rft.place=Essex&rft.pub=Longman+Scientific+and+Technical&rft.date=1978&rft.isbn=0-582-44194-3&rft.aulast=Foster&rft.aufirst=J.&rft.au=Nightingale%2C+J.D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fshortcourseingen0000fost&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleppnerKolenkow1978" class="citation book cs1"><a href="/wiki/Daniel_Kleppner" title="Daniel Kleppner">Kleppner, D.</a>; Kolenkow, R.J. (1978). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontome00dani"><i>An introduction to mechanics</i></a>. <a href="/wiki/McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-035048-5" title="Special:BookSources/0-07-035048-5"><bdi>0-07-035048-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+mechanics&rft.pub=McGraw%E2%80%93Hill&rft.date=1978&rft.isbn=0-07-035048-5&rft.aulast=Kleppner&rft.aufirst=D.&rft.au=Kolenkow%2C+R.J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontome00dani&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKopeikinEfroimskyKaplan2011" class="citation book cs1">Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RfR2GawB-xcC"><i>Relativistic Celestial Mechanics of the Solar System</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-527-40856-6" title="Special:BookSources/978-3-527-40856-6"><bdi>978-3-527-40856-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Relativistic+Celestial+Mechanics+of+the+Solar+System&rft.pub=John+Wiley+%26+Sons&rft.date=2011&rft.isbn=978-3-527-40856-6&rft.aulast=Kopeikin&rft.aufirst=Sergei&rft.au=Efroimsky%2C+Michael&rft.au=Kaplan%2C+George&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRfR2GawB-xcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandauLifshitz1975" class="citation book cs1"><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, L. D.</a>; <a href="/wiki/Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a> (1975). <i>The classical theory of fields</i>. Course of theoretical physics. Vol. 2 (4th ed.). Oxford: <a href="/wiki/Butterworth%E2%80%93Heinemann" class="mw-redirect" title="Butterworth–Heinemann">Butterworth–Heinemann</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7506-2768-9" title="Special:BookSources/0-7506-2768-9"><bdi>0-7506-2768-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+classical+theory+of+fields&rft.place=Oxford&rft.series=Course+of+theoretical+physics&rft.edition=4th&rft.pub=Butterworth%E2%80%93Heinemann&rft.date=1975&rft.isbn=0-7506-2768-9&rft.aulast=Landau&rft.aufirst=L.+D.&rft.au=Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawden2012" class="citation book cs1">Lawden, Derek F. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MSBB_ENAhT4C"><i>An Introduction to Tensor Calculus: Relativity and Cosmology</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13214-3" title="Special:BookSources/978-0-486-13214-3"><bdi>978-0-486-13214-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Tensor+Calculus%3A+Relativity+and+Cosmology&rft.pub=Courier+Corporation&rft.date=2012&rft.isbn=978-0-486-13214-3&rft.aulast=Lawden&rft.aufirst=Derek+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DMSBB_ENAhT4C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLovelockRund1989" class="citation cs2"><a href="/wiki/David_Lovelock" title="David Lovelock">Lovelock, David</a>; <a href="/wiki/Hanno_Rund" title="Hanno Rund">Rund, Hanno</a> (1989), <i>Tensors, Differential Forms, and Variational Principles</i>, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65840-6" title="Special:BookSources/0-486-65840-6"><bdi>0-486-65840-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensors%2C+Differential+Forms%2C+and+Variational+Principles&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1989&rft.isbn=0-486-65840-6&rft.aulast=Lovelock&rft.aufirst=David&rft.au=Rund%2C+Hanno&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMinkowski1908" class="citation cs2"><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski, Hermann</a> (1908), <a rel="nofollow" class="external text" href="https://archive.today/20120708062436/http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=62931">"Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern"</a>, <i>Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-August-Universität zu Göttingen</i>, Göttingen, archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/no_cache/en/dms/load/img/?IDDOC=62931">the original</a> on 2012-07-08</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nachrichten+von+der+K%C3%B6niglichen+Gesellschaft+der+Wissenschaften+und+der+Georg-August-Universit%C3%A4t+zu+G%C3%B6ttingen&rft.atitle=Die+Grundgleichungen+f%C3%BCr+die+elektromagnetischen+Vorg%C3%A4nge+in+bewegten+K%C3%B6rpern&rft.date=1908&rft.aulast=Minkowski&rft.aufirst=Hermann&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Fno_cache%2Fen%2Fdms%2Fload%2Fimg%2F%3FIDDOC%3D62931&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoisson2004" class="citation cs2"><a href="/wiki/Eric_Poisson" title="Eric Poisson">Poisson, Eric</a> (2004), <i>A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0521537803" title="Special:BookSources/978-0521537803"><bdi>978-0521537803</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Relativist%27s+Toolkit%3A+The+Mathematics+of+Black-Hole+Mechanics&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=978-0521537803&rft.aulast=Poisson&rft.aufirst=Eric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeinberg1972" class="citation cs2"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1972), <a rel="nofollow" class="external text" href="https://archive.org/details/gravitationcosmo00stev_0"><i>Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity</i></a>, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92567-5" title="Special:BookSources/978-0-471-92567-5"><bdi>978-0-471-92567-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Cosmology%3A+Principles+and+Applications+of+the+General+Theory+of+Relativity&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=978-0-471-92567-5&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgravitationcosmo00stev_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZwiebach2004" class="citation book cs1"><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach, Barton</a> (2004). <i>A First Course in String Theory</i> (first ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83143-1" title="Special:BookSources/0-521-83143-1"><bdi>0-521-83143-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+String+Theory&rft.edition=first&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0-521-83143-1&rft.aulast=Zwiebach&rft.aufirst=Barton&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProper+time" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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href="/wiki/Present" title="Present">Present</a></li> <li><a href="/wiki/Future" title="Future">Future</a></li> <li><a href="/wiki/Eternity" title="Eternity">Eternity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Measurement</a><br />and <a href="/wiki/Time_standard" title="Time standard">standards</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinated_Universal_Time" title="Coordinated Universal Time">UTC</a></li> <li><a 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href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Relative_hour" title="Relative hour">Relative hour</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Traditional_Chinese_timekeeping" title="Traditional Chinese timekeeping">Chinese</a></li> <li><a href="/wiki/Decimal_time" title="Decimal time">Decimal</a></li> <li><a href="/wiki/Hexadecimal_time" title="Hexadecimal time">Hexadecimal</a></li> <li><a href="/wiki/Hindu_units_of_time" title="Hindu units of time">Hindu</a></li> <li><a href="/wiki/Metric_time" title="Metric time">Metric</a></li> <li><a href="/wiki/Roman_timekeeping" title="Roman timekeeping">Roman</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Calendar" title="Calendar">Calendars</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Calendar#Systems" title="Calendar">Main types</a> <ul><li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li></ul></li> <li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Chinese_calendar" title="Chinese calendar">Chinese</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu Panchang</a></li> <li><a href="/wiki/Maya_calendar" title="Maya calendar">Maya</a></li> <li><i><a href="/wiki/List_of_calendars" title="List of calendars">List</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Clock" title="Clock">Clocks</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clock#Types" title="Clock">Main types</a> <ul><li><a href="/wiki/Astronomical_clock" title="Astronomical clock">astronomical</a> <ul><li><a href="/wiki/Astrarium" title="Astrarium">astrarium</a></li></ul></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">atomic</a> <ul><li><a href="/wiki/Quantum_clock" class="mw-redirect" title="Quantum clock">quantum</a></li></ul></li> <li><a href="/wiki/Hourglass" title="Hourglass">hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">marine</a></li> <li><a href="/wiki/Sundial" title="Sundial">sundial</a></li> <li><a href="/wiki/Watch" title="Watch">watch</a> <ul><li><a href="/wiki/Mechanical_watch" title="Mechanical watch">mechanical</a></li> <li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">water-based</a></li></ul></li> <li><a href="/wiki/Cuckoo_clock" title="Cuckoo clock">Cuckoo clock</a></li> <li><a href="/wiki/Digital_clock" title="Digital clock">Digital clock</a></li> <li><a href="/wiki/Grandfather_clock" title="Grandfather clock">Grandfather clock</a></li> <li><i><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History</a></i> <ul><li><i><a href="/wiki/Timeline_of_time_measurement_inventions" title="Timeline of time measurement inventions">Timeline</a></i></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Chronology" title="Chronology">Chronology</a></li><li><a href="/wiki/History" title="History">History</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></li> <li><a href="/wiki/Big_History" title="Big History">Big History</a></li> <li><a href="/wiki/Calendar_era" title="Calendar era">Calendar era</a></li> <li><a href="/wiki/Deep_time" title="Deep time">Deep time</a></li> <li><a href="/wiki/Periodization" title="Periodization">Periodization</a></li> <li><a href="/wiki/Regnal_year" title="Regnal year">Regnal year</a></li> <li><a href="/wiki/Timeline" title="Timeline">Timeline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Philosophy_of_space_and_time" title="Philosophy of space and time">Philosophy of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/A_series_and_B_series" title="A series and B series">A series and B series</a></li> <li><a href="/wiki/B-theory_of_time" title="B-theory of time">B-theory of time</a></li> <li><a href="/wiki/Chronocentrism" title="Chronocentrism">Chronocentrism</a></li> <li><a href="/wiki/Duration_(philosophy)" title="Duration (philosophy)">Duration</a></li> <li><a href="/wiki/Endurantism" title="Endurantism">Endurantism</a></li> <li><a href="/wiki/Eternal_return" title="Eternal return">Eternal return</a></li> <li><a href="/wiki/Eternalism_(philosophy_of_time)" title="Eternalism (philosophy of time)">Eternalism</a></li> <li><a href="/wiki/Event_(philosophy)" title="Event (philosophy)">Event</a></li> <li><a href="/wiki/Perdurantism" title="Perdurantism">Perdurantism</a></li> <li><a href="/wiki/Philosophical_presentism" title="Philosophical presentism">Presentism</a></li> <li><a href="/wiki/Temporal_finitism" title="Temporal finitism">Temporal finitism</a></li> <li><a href="/wiki/Temporal_parts" title="Temporal parts">Temporal parts</a></li> <li><i><a href="/wiki/The_Unreality_of_Time" title="The Unreality of Time">The Unreality of Time</a></i></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="/wiki/Category:Time_in_religion" title="Category:Time in religion">Religion</a></li><li><a href="/wiki/Template:Time_in_religion_and_mythology" title="Template:Time in religion and mythology">Mythology</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ages_of_Man" title="Ages of Man">Ages of Man</a></li> <li><a href="/wiki/Destiny" title="Destiny">Destiny</a></li> <li><a href="/wiki/Immortality" title="Immortality">Immortality</a></li> <li><a href="/wiki/The_Dreaming" title="The Dreaming">Dreamtime</a></li> <li><a href="/wiki/K%C4%81la" title="Kāla">Kāla</a></li> <li><a href="/wiki/Time_and_fate_deities" title="Time and fate deities">Time and fate deities</a> <ul><li><a href="/wiki/Father_Time" title="Father Time">Father Time</a></li></ul></li> <li><a href="/wiki/Wheel_of_time" title="Wheel of time">Wheel of time</a> <ul><li><a href="/wiki/Kalachakra" title="Kalachakra">Kalachakra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_perception" title="Time perception">Human experience</a><br />and <a href="/wiki/Time-use_research" title="Time-use research">use of time</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronemics" title="Chronemics">Chronemics</a></li> <li><a href="/wiki/Generation_time" title="Generation time">Generation time</a></li> <li><a href="/wiki/Mental_chronometry" title="Mental chronometry">Mental chronometry</a></li> <li><a href="/wiki/Duration_(music)" title="Duration (music)">Music</a> <ul><li><a href="/wiki/Tempo" title="Tempo">tempo</a></li> <li><a href="/wiki/Time_signature" title="Time signature">time signature</a></li></ul></li> <li><a href="/wiki/Rosy_retrospection" title="Rosy retrospection">Rosy retrospection</a></li> <li><a href="/wiki/Tense%E2%80%93aspect%E2%80%93mood" title="Tense–aspect–mood">Tense–aspect–mood</a></li> <li><a href="/wiki/Time_management" title="Time management">Time management</a></li> <li><a href="/wiki/Yesterday_(time)" title="Yesterday (time)">Yesterday</a> – <a href="/wiki/Present" title="Present">Today</a> – <a href="/wiki/Tomorrow_(time)" title="Tomorrow (time)">Tomorrow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Time in <a href="/wiki/Science" title="Science">science</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Geology" title="Geology">Geology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geologic_time_scale" title="Geologic time scale">Geological time</a> <ul><li><a href="/wiki/Age_(geology)" class="mw-redirect" title="Age (geology)">age</a></li> <li><a href="/wiki/Chronozone" title="Chronozone">chron</a></li> <li><a href="/wiki/Eon_(geology)" class="mw-redirect" title="Eon (geology)">eon</a></li> <li><a href="/wiki/Epoch_(geology)" class="mw-redirect" title="Epoch (geology)">epoch</a></li> <li><a href="/wiki/Era_(geology)" class="mw-redirect" title="Era (geology)">era</a></li> <li><a href="/wiki/Geological_period" class="mw-redirect" title="Geological period">period</a></li></ul></li> <li><a href="/wiki/Geochronology" title="Geochronology">Geochronology</a></li> <li><a href="/wiki/Geological_history_of_Earth" title="Geological history of Earth">Geological history of Earth</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;"><a href="/wiki/Time_in_physics" title="Time in physics">Physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Arrow_of_time" title="Arrow of time">Arrow of time</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a class="mw-selflink selflink">Proper time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time_translation_symmetry" class="mw-redirect" title="Time translation symmetry">Time translation symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">Time reversal symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:6.5em;font-weight:normal; text-align:center;">Other fields</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Chronobiology" title="Chronobiology">Chronobiology</a> <ul><li><a href="/wiki/Circadian_rhythm" title="Circadian rhythm">Circadian rhythms</a></li></ul></li> <li><a href="/wiki/Chemical_clock" title="Chemical clock">Clock reaction</a></li> <li><a href="/wiki/Glottochronology" title="Glottochronology">Glottochronology</a></li> <li><a href="/wiki/Time_geography" title="Time geography">Time geography</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Memory" title="Memory">Memory</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Space" title="Space">Space</a></li> <li><a href="/wiki/System_time" title="System time">System time</a></li> <li><i><a href="/wiki/Tempus_fugit" title="Tempus fugit">Tempus fugit</a></i></li> <li><a href="/wiki/Time_capsule" title="Time capsule">Time capsule</a></li> <li><a href="/wiki/Time_immemorial" title="Time immemorial">Time immemorial</a></li> <li><a href="/wiki/Time_travel" title="Time travel">Time travel</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Time" title="Category:Time">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Commons page"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> <a href="https://commons.wikimedia.org/wiki/Category:Time" class="extiw" title="commons:Category:Time">Commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Time_measurement_and_standards342" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Time_measurement_and_standards" title="Template:Time measurement and standards"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Time_measurement_and_standards" title="Template talk:Time measurement and standards"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Time_measurement_and_standards" title="Special:EditPage/Template:Time measurement and standards"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Time_measurement_and_standards342" style="font-size:114%;margin:0 4em"><a href="/wiki/Time" title="Time">Time measurement</a> and <a href="/wiki/Time_standard" title="Time standard">standards</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Chronometry" title="Chronometry">Chronometry</a></li> <li><a href="/wiki/Orders_of_magnitude_(time)" title="Orders of magnitude (time)">Orders of magnitude</a></li> <li><a href="/wiki/Time_metrology" class="mw-redirect" title="Time metrology">Metrology</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">International standards</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Coordinated_Universal_Time" title="Coordinated Universal Time">Coordinated Universal Time</a> <ul><li><a href="/wiki/UTC_offset" title="UTC offset">offset</a></li></ul></li> <li><a href="/wiki/Universal_Time" title="Universal Time">UT</a></li> <li><a href="/wiki/%CE%94T_(timekeeping)" title="ΔT (timekeeping)">ΔT</a></li> <li><a href="/wiki/DUT1" title="DUT1">DUT1</a></li> <li><a href="/wiki/International_Earth_Rotation_and_Reference_Systems_Service" title="International Earth Rotation and Reference Systems Service">International Earth Rotation and Reference Systems Service</a></li> <li><a href="/wiki/ISO_31-1" title="ISO 31-1">ISO 31-1</a></li> <li><a href="/wiki/ISO_8601" title="ISO 8601">ISO 8601</a></li> <li><a href="/wiki/International_Atomic_Time" title="International Atomic Time">International Atomic Time</a></li> <li><a href="/wiki/12-hour_clock" title="12-hour clock">12-hour clock</a></li> <li><a href="/wiki/24-hour_clock" title="24-hour clock">24-hour clock</a></li> <li><a href="/wiki/Barycentric_Coordinate_Time" title="Barycentric Coordinate Time">Barycentric Coordinate Time</a></li> <li><a href="/wiki/Barycentric_Dynamical_Time" title="Barycentric Dynamical Time">Barycentric Dynamical Time</a></li> <li><a href="/wiki/Civil_time" title="Civil time">Civil time</a></li> <li><a href="/wiki/Daylight_saving_time" title="Daylight saving time">Daylight saving time</a></li> <li><a href="/wiki/Geocentric_Coordinate_Time" title="Geocentric Coordinate Time">Geocentric Coordinate Time</a></li> <li><a href="/wiki/International_Date_Line" title="International Date Line">International Date Line</a></li> <li><a href="/wiki/IERS_Reference_Meridian" title="IERS Reference Meridian">IERS Reference Meridian</a></li> <li><a href="/wiki/Leap_second" title="Leap second">Leap second</a></li> <li><a href="/wiki/Solar_time" title="Solar time">Solar time</a></li> <li><a href="/wiki/Terrestrial_Time" title="Terrestrial Time">Terrestrial Time</a></li> <li><a href="/wiki/Time_zone" title="Time zone">Time zone</a></li> <li><a href="/wiki/180th_meridian" title="180th meridian">180th meridian</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="9" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Hourglass" title="Hourglass"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/75px-Marine_sandglass_MMM.jpg" decoding="async" width="75" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/113px-Marine_sandglass_MMM.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/70/Marine_sandglass_MMM.jpg/150px-Marine_sandglass_MMM.jpg 2x" data-file-width="1637" data-file-height="3819" /></a></span><br /><span typeof="mw:File"><a href="/wiki/Time_zone" title="Time zone"><img alt="template illustration" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/75px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png" decoding="async" width="75" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/113px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png/150px-Aleutian_Islands_with_180th_meridian_and_International_Date_Line_%28cropped%29.png 2x" data-file-width="496" data-file-height="1007" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Obsolete standards</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ephemeris_time" title="Ephemeris time">Ephemeris time</a></li> <li><a href="/wiki/Greenwich_Mean_Time" title="Greenwich Mean Time">Greenwich Mean Time</a></li> <li><a href="/wiki/Prime_meridian" title="Prime meridian">Prime meridian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Time_in_physics" title="Time in physics">Time in physics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</a></li> <li><a href="/wiki/Spacetime" title="Spacetime">Spacetime</a></li> <li><a href="/wiki/Chronon" title="Chronon">Chronon</a></li> <li><a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">Continuous signal</a></li> <li><a href="/wiki/Coordinate_time" title="Coordinate time">Coordinate time</a></li> <li><a href="/wiki/Cosmological_decade" class="mw-redirect" title="Cosmological decade">Cosmological decade</a></li> <li><a href="/wiki/Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete time and continuous time</a></li> <li><a class="mw-selflink selflink">Proper time</a></li> <li><a href="/wiki/Theory_of_relativity" title="Theory of relativity">Theory of relativity</a></li> <li><a href="/wiki/Time_dilation" title="Time dilation">Time dilation</a></li> <li><a href="/wiki/Gravitational_time_dilation" title="Gravitational time dilation">Gravitational time dilation</a></li> <li><a href="/wiki/Time_domain" title="Time domain">Time domain</a></li> <li><a href="/wiki/Time-translation_symmetry" title="Time-translation symmetry">Time-translation symmetry</a></li> <li><a href="/wiki/T-symmetry" title="T-symmetry">T-symmetry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Horology" class="mw-redirect" title="Horology">Horology</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clock" title="Clock">Clock</a></li> <li><a href="/wiki/Astrarium" title="Astrarium">Astrarium</a></li> <li><a href="/wiki/Atomic_clock" title="Atomic clock">Atomic clock</a></li> <li><a href="/wiki/Complication_(horology)" title="Complication (horology)">Complication</a></li> <li><a href="/wiki/History_of_timekeeping_devices" title="History of timekeeping devices">History of timekeeping devices</a></li> <li><a href="/wiki/Hourglass" title="Hourglass">Hourglass</a></li> <li><a href="/wiki/Marine_chronometer" title="Marine chronometer">Marine chronometer</a></li> <li><a href="/wiki/Marine_sandglass" title="Marine sandglass">Marine sandglass</a></li> <li><a href="/wiki/Radio_clock" title="Radio clock">Radio clock</a></li> <li><a href="/wiki/Watch" title="Watch">Watch</a> <ul><li><a href="/wiki/Stopwatch" title="Stopwatch">stopwatch</a></li></ul></li> <li><a href="/wiki/Water_clock" title="Water clock">Water clock</a></li> <li><a href="/wiki/Sundial" title="Sundial">Sundial</a></li> <li><a href="/wiki/Dialing_scales" title="Dialing scales">Dialing scales</a></li> <li><a href="/wiki/Equation_of_time" title="Equation of time">Equation of time</a></li> <li><a href="/wiki/History_of_sundials" title="History of sundials">History of sundials</a></li> <li><a href="/wiki/Schema_for_horizontal_dials" title="Schema for horizontal dials">Sundial markup schema</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Calendar" title="Calendar">Calendar</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gregorian_calendar" title="Gregorian calendar">Gregorian</a></li> <li><a href="/wiki/Hebrew_calendar" title="Hebrew calendar">Hebrew</a></li> <li><a href="/wiki/Hindu_calendar" title="Hindu calendar">Hindu</a></li> <li><a href="/wiki/Holocene_calendar" title="Holocene calendar">Holocene</a></li> <li><a href="/wiki/Islamic_calendar" title="Islamic calendar">Islamic</a> (lunar Hijri)</li> <li><a href="/wiki/Julian_calendar" title="Julian calendar">Julian</a></li> <li><a href="/wiki/Solar_Hijri_calendar" title="Solar Hijri calendar">Solar Hijri</a></li> <li><a href="/wiki/Astronomical_year_numbering" title="Astronomical year numbering">Astronomical</a></li> <li><a href="/wiki/Dominical_letter" title="Dominical letter">Dominical letter</a></li> <li><a href="/wiki/Epact" title="Epact">Epact</a></li> <li><a href="/wiki/Equinox" title="Equinox">Equinox</a></li> <li><a href="/wiki/Intercalation_(timekeeping)" title="Intercalation (timekeeping)">Intercalation</a></li> <li><a href="/wiki/Julian_day" title="Julian day">Julian day</a></li> <li><a href="/wiki/Leap_year" title="Leap year">Leap year</a></li> <li><a href="/wiki/Lunar_calendar" title="Lunar calendar">Lunar</a></li> <li><a href="/wiki/Lunisolar_calendar" title="Lunisolar calendar">Lunisolar</a></li> <li><a href="/wiki/Solar_calendar" title="Solar calendar">Solar</a></li> <li><a href="/wiki/Solstice" title="Solstice">Solstice</a></li> <li><a href="/wiki/Tropical_year" title="Tropical year">Tropical year</a></li> <li><a href="/wiki/Determination_of_the_day_of_the_week" title="Determination of the day of the week">Weekday determination</a></li> <li><a href="/wiki/Names_of_the_days_of_the_week" title="Names of the days of the week">Weekday names</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Archaeology and geology</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Chronological_dating" title="Chronological dating">Chronological dating</a></li> <li><a href="/wiki/Geologic_time_scale" title="Geologic time scale">Geologic time scale</a></li> <li><a href="/wiki/International_Commission_on_Stratigraphy" title="International Commission on Stratigraphy">International Commission on Stratigraphy</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Astronomical_chronology" title="Astronomical chronology">Astronomical chronology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Galactic_year" title="Galactic year">Galactic year</a></li> <li><a href="/wiki/Nuclear_timescale" title="Nuclear timescale">Nuclear timescale</a></li> <li><a href="/wiki/Precession" title="Precession">Precession</a></li> <li><a href="/wiki/Sidereal_time" title="Sidereal time">Sidereal time</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Unit_of_time" title="Unit of time">units of time</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Instant" title="Instant">Instant</a></li> <li><a href="/wiki/Flick_(time)" title="Flick (time)">Flick</a></li> <li><a href="/wiki/Shake_(unit)" title="Shake (unit)">Shake</a></li> <li><a href="/wiki/Jiffy_(time)" title="Jiffy (time)">Jiffy</a></li> <li><a href="/wiki/Second" title="Second">Second</a></li> <li><a href="/wiki/Minute" title="Minute">Minute</a></li> <li><a href="/wiki/Moment_(unit)" title="Moment (unit)">Moment</a></li> <li><a href="/wiki/Hour" title="Hour">Hour</a></li> <li><a href="/wiki/Day" title="Day">Day</a></li> <li><a href="/wiki/Week" title="Week">Week</a></li> <li><a href="/wiki/Fortnight" title="Fortnight">Fortnight</a></li> <li><a href="/wiki/Month" title="Month">Month</a></li> <li><a href="/wiki/Year" title="Year">Year</a></li> <li><a href="/wiki/Olympiad" title="Olympiad">Olympiad</a></li> 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Proper time - Wikipedia